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Math and science::Algebra::Aluffi

Groups. Some properties of order.

Property 1

Suppose that \( g^2 = e \) for all elements \( g \) in group \( G \). Then we can say that \( G \) is commutative.

Property 2

Suppose \( g \) is an element with odd order. Then \( |g^2| = |g| \)?

Property 3

The order of \( [m]_n \) in \( \mathbb{Z}/n\mathbb{Z} \) is 1 if \( n | m \), and more generally:

\[ | [m]_n| = \frac{n}{\operatorname{gcd}(m, n)} \]

Property 4

Following from property 3:

  • The class \( [m]_n \) generates \( \mathbb{Z}/nZ \) iff \( \operatorname{gcd}(m, n) = 1 \).
  • the order of every element of \( \mathbb{Z}/n\mathbb{Z} \) divides \( n \), which is the order, \( |\mathbb{Z}/n\mathbb{Z}| \).

TODO: proof
TODO: check if the properties are actually correct.

Source

Aluffi, p48
Aluffi, p49
Aluffi, p55